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How could I calculate the total 𝛥𝑣 required to change the inclination of a Geostationary orbit to another circular orbit with an inclination of 30° and the same radius? (The sidereal day is used for Geostationary orbit calculation.)

Currently my idea is to solve it from the formula of inclination, $$\arccos\left(\frac{hk}{|h|}\right)$$ where $h = rv$ and the change from zero to 30 degrees means the component of $h$ on the $k$ axis is increasing.

Since $r$ does not change and $hk$ is changing, the velocity $v$ should change. I'm not too sure what to do next.

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  • $\begingroup$ My idea now is to solve it from the formula of inclination, arccos(hk/|h|). where h = r*v. And the change from 0 degrees to 30 degrees means the component of h on the k axis is increasing. Since r does not change, and hk is changing, the velocity v should change. I'm not too sure what to do next. $\endgroup$ – Yibowen Zhao Nov 13 '19 at 16:47
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    $\begingroup$ en.m.wikipedia.org/wiki/Orbital_inclination_change $\endgroup$ – Russell Borogove Nov 13 '19 at 17:23
  • $\begingroup$ Did you just want the equation, or were you also interested in how the equation is derived? $\endgroup$ – uhoh Nov 13 '19 at 17:56
  • $\begingroup$ I just want the equation, thanks $\endgroup$ – Yibowen Zhao Nov 14 '19 at 7:14
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According to Wikipedia, the general equation for inclination change is:

$$\Delta{v_i}= {2\sin(\frac{\Delta{i}}{2})\sqrt{1-e^2}\cos(\omega+f)na \over {(1+e\cos(f))}}$$

Where:

  • $e\,$ is the orbital eccentricity
  • $\omega\,$ is the argument of periapsis
  • $f\,$ is the true anomaly
  • $n\,$ is the mean motion
  • $a\,$ is the semi-major axis

For circular orbits, this simplifies considerably to:

$$\Delta{v_i}= {2v\, \sin \left(\frac{\Delta{i}}{2} \right)}$$

Where $v\,$ is the orbital velocity and has the same units as $\Delta{v_i}$.

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